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Following the recent breakdown in diplomatic relations, the government of Honestants and Swindlecants is taking a hard line on foreigners, declaring all “Normaltons” (people who can tell the truth or lie at will) detrimental to the logical integrity of society. Yesterday I was backpacking on the island with a companion. The secret police found us and took us both to a detention centre. We were forcibly separated and I was locked in a solitary cell. That night I heard terrible screams, and in the morning there was an eerie silence. Fearing the worst, I called out my companion’s name, but there was no reply. There were two guards outside my cell so I asked them what had become of my companion.

Guard 1: “If your companion is dead, the other guard is an Honestant.”

Guard 2: “If the other guard is an Honestant, your companion is alive. Unfortunately, the other guard is a Swindlecant.”

That sounds like bad news, but is it really? Is my companion alive, or dead?

g2=H makes his conclusion true [-> companion alive] OR his premise false [-> g1=S]. Since g2=H implies g1=S, the OR is satisfied for either state of the companion. That's consistent, but it gives no information about the companion. g1=S makes g1's premise true [-> companion alive] and conclusion false [-> g2=S]. That's a contradiction.

There seems to be no self-consistent assignment of S and H for g1 and g2.

I came to a similar dead end assuming [1] companion was alive -> contradiction and [2] companion was dead -> contradiction.

For an inference If A then B I use the truth values of ^A or B: if the premise is false or the conclusion is true, than the inference is true. That is, an inference is false if and only if A is true and B is false.

Guard 1: “If your companion is dead, the other guard is an Honestant.”

Guard 2: “If the other guard is an Honestant, your companion is alive. Unfortunately, the other guard is a Swindlecant.”

possibilities:A=aliveD=deadH=honestantS=swindelcant

A-HHA-HSA-SHA-SSD-HHD-HSD-SHD-SS

* take AHH, if they are both honestants, then both are lying. Cross AHH out* take AHS, the first guy is telling the truth but then the second guy WOULDNT be lying. Cross AHS out* take ASH, the first guy's statement works if the second guy is the honestant. But the second guy's first statement is a contradiction. Cross ASH out* take a**, the first guy's statement only works if your companion dies, so this is already a contradiction. Cross a** out

we've eliminated all the alive possibilities

your companion is dead

am I right?

just in case, I will look at the rest:

DHH- the second guy is lying, contradiction. Cross DHH outDHS- this fits as far as I can tellDSH- contradiction, cross it outDSS- guard 2 is telling the truth, contradiction

* take AHH, if they are both honestants, then both are lying. Cross AHH out* take AHS, the first guy is telling the truth but then the second guy WOULDNT be lying. Cross AHS out* take ASH, the first guy's statement works if the second guy is the honestant. But the second guy's first statement is a contradiction. Cross ASH out* take a**, the first guy's statement only works if your companion dies, so this is already a contradiction. Cross a** out

we've eliminated all the alive possibilities

your companion is dead

am I right?

just in case, I will look at the rest:

DHH- the second guy is lying, contradiction. Cross DHH outDHS- this fits as far as I can tellDSH- contradiction, cross it outDSS- guard 2 is telling the truth, contradiction

leaving

DHS

your companion is deadguard 1: Honestantguard 2: swindelcant

Spoiler for DHS doen't really work

D = Friend DeadH = Guard 1 HonestantS = Guard 2 Swindlecant

But Guard 1 lied in that scenario: If Friend dead (True) then Guard 2 Honestant (False) = FalseSo he cannot be a Honestant!